Infinitesimal rotation of classical fields: why are rotation group representations important?

Question

I understand that $SO(3)$ representations are important in quantum physics, because eigenspaces of the Hamiltonian are irreps of $SO(3)$ if it is part of the symmetry group. But I don't see the reason why the irreps should be essential for classical field theories.

My professor talked about the conservation of angular momentum because of Noether's theorem. The infinitesimal symmetry is

$$ x^\mu \mapsto x^\mu + \omega^{\mu\nu}x_\nu $$

with an asymmetric matrix $\omega$, so the field transforms as

$$ \phi_i(x) \mapsto {S_i}^j(\omega)\phi_j(x^\mu + \omega^{\mu\nu}x_\nu) $$

where $S$ is the representation of $SO(3)$, which is trivial for scalar fields.

• Why are representations necessary for such an infinitesimal symmetry? For translations, the $x^\mu$ transformation was sufficient, no talk about representations.
• If the trivial irrep is for scalars, are all the other (real) irreps for vectors, and their tensor products for higher tensors?

Answer 1

No one talked about representations for the translations because the translations do not change the natural basis for the vectors. When you apply a rotation, you are rotating the natural coordinate system your vectors and tensors are represented in, so fields which are components of those have to change. Translations don't change the natural coordinate basis at a point, so they don't affect such fields. In other words, no one talked about representations of the translation group because every field you normally think of transforms in the trivial representation.

The lowest non-trivial irreducible representation (labelled as "spin-1" in the usual diction) of $\mathrm{SO}(3)$, which is its fundamental representation, is the vector representation (let's call its representation space $V$). The tensor representations of rank $(k,l)$ are now $V\otimes\dots\otimes V\otimes V^*\otimes\dots\otimes V^*$ with $k$ factors of $V$ and $l$ factors of $V^*$. They are not irreducible, and split at least into the trace, symmetric traceless tensor and antisymmetric tensor representations.

The irreducible representations of higher spin do not usually play a special role, but they are not for "vectors"! For instance, the "next" representation labelled as "spin-2" is five-dimensional, and is that of the symmetric traceless matrices/rank-2 tensors.